sciencefandomcom_el-20200214-history
Εξίσωση Ορμής Cauchy
Εξίσωση Ορμής Equation thumb|300px| [[Επιστημονικός Νόμος Επιστημονικοί Νόμοι ---- Μαθηματικό Θεώρημα Νόμοι Μαθηματικών ---- Φυσικός Νόμος Νόμοι Φυσικής ---- Νόμοι Χημείας ---- Νόμοι Γεωλογίας ---- Νόμοι Βιολογίας ---- Νόμοι Οικονομίας ]] thumb|300px| [[Επιστήμη Επιστήμες Φυσικές Επιστήμες Βιο-Επιστήμες Γεω-Επιστήμες Οικονομικές Επιστήμες Θεωρητικές Επιστήμες Κοινωνικές Επιστήμες Επιστήμες Υγείας ---- Τεχνολογία ---- Επιστημονικός Κλάδος Επιστημονικός Νόμος Επιστημονική Μέθοδος Επιστημονική Θεωρία Επιστημονικά Κέντρα Γης Επιστήμονες Γης ]] - Νόμος της Φυσικής. - Ακριβέστερα, είναι ένας νόμος της Ρευστομηχανικής - Χρονολογία ανακάλυψης: Ετυμολογία Η ονομασία "νόμος" σχετίζεται ετυμολογικά με το όνομα του φυσικού επιστήμονα " ". Διατύπωση The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum: : \rho \frac{D \mathbf{v}}{D t} = \nabla \cdot \boldsymbol{\sigma} + \mathbf{f} or, with the derivative expanded out, : \rho \left\mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v}\right = \nabla \cdot \boldsymbol{\sigma} + \mathbf{f} where \rho is the density of the continuum, \boldsymbol{\sigma} is the stress tensor, and \mathbf{f} contains all of the body forces (normally just gravity). \mathbf{v} is the velocity vector field, which depends on time and space. The stress tensor is sometimes split into pressure and the deviatoric stress tensor: : \boldsymbol{\sigma} = -p\mathbb{I} + \mathbb{T} where \scriptstyle \mathbb{I} is the \scriptstyle 3 \times 3 identity matrix and \scriptstyle \mathbb{T} the deviatoric stress tensor. The divergence of the stress tensor can be written as : \nabla \cdot \boldsymbol{\sigma} = -\nabla p + \nabla \cdot\mathbb{T}. All non-relativistic momentum conservation equations, such as the Navier-Stokes equation, can be derived by beginning with the Cauchy momentum equation and specifying the stress tensor through a constitutive relation. Derivation Applying Newton's second law ( i^{th} component) to a control volume in the continuum being modeled gives: : m a_i = F_i\, : \rho \int_{\Omega} \frac{d v_i}{d t} \, dV = \int_{\Omega} \nabla_j\sigma_i^j \, dV + \int_{\Omega} f_i \, dV : \int_{\Omega} (\rho \frac{d v_i}{d t} - \nabla_j\sigma_i^j - f_i )\, dV = 0 : \rho \dot{v_i} - \nabla_j\sigma_i^j - f_i = 0 where \Omega represents the control volume. Since this equation must hold for any control volume, it must be true that the integrand is zero, from this the Cauchy momentum equation follows. The main challenge in deriving this equation is establishing that the derivative of the stress tensor is one of the forces that constitutes F_i . Cartesian coordinates : \rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}+ w \frac{\partial u}{\partial z}\right) = -\frac{\partial P}{\partial x} - \frac{\partial \tau_{xx}}{\partial x} - \frac{\partial \tau_{yx}}{\partial y} - \frac{\partial \tau_{zx}}{\partial z} + \rho g_x : \rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y}+ w \frac{\partial v}{\partial z}\right) = -\frac{\partial P}{\partial y} - \frac{\partial \tau_{yy}}{\partial x} - \frac{\partial \tau_{xy}}{\partial y} - \frac{\partial \tau_{zy}}{\partial z} + \rho g_y : \rho \left(\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y}+ w \frac{\partial w}{\partial z}\right) = - \frac{\partial P}{\partial z} - \frac{\partial \tau_{zz}}{\partial x} - \frac{\partial \tau_{yz}}{\partial y} - \frac{\partial \tau_{xz}}{\partial z} + \rho g_z. Cylindrical coordinates : r:\;\;\rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_r}{\partial \phi} + u_z \frac{\partial u_r}{\partial z} - \frac{u_{\phi}^2}{r}\right) = -\frac{\partial P}{\partial r} - \frac{1}{r}\frac{\partial {(r{\tau_{rr})}}}{\partial r} - \frac{1}{r}\frac{\partial {\tau_{\phi r}}}{\partial \phi} - \frac{\partial {\tau_{z r}}}{\partial z} + \frac {\tau_{\phi \phi}}{r} + \rho g_r : \phi:\;\;\rho \left(\frac{\partial u_{\phi}}{\partial t} + u_r \frac{\partial u_{\phi}}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_{\phi}}{\partial \phi} + u_z \frac{\partial u_{\phi}}{\partial z} + \frac{u_r u_{\phi}}{r}\right) = -\frac{1}{r}\frac{\partial P}{\partial \phi} -\frac{1}{r}\frac{\partial {\tau_{\phi \phi}}}{\partial \phi} - \frac{1}{r^2}\frac{\partial {(r^2{\tau_{r \phi})}}}{\partial r} - \frac{\partial {\tau_{z r}}}{\partial z} + \rho g_{\phi} : z:\;\;\rho \left(\frac{\partial u_z}{\partial t} + u_r \frac{\partial u_z}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_z}{\partial \phi} + u_z \frac{\partial u_z}{\partial z}\right) = -\frac{\partial P}{\partial z} - \frac{\partial {\tau_{z z}}}{\partial z} - \frac{1}{r}\frac{\partial {\tau_{\phi z}}}{\partial \phi} - \frac{1}{r}\frac{\partial {(r{\tau_{rz})}}}{\partial r} + \rho g_z. By expressing the shear stress in terms of viscosity and fluid velocity, and assuming constant density and viscosity, the Cauchy momentum equation will lead to the Navier–Stokes equations. By assuming inviscid flow, the Navier–Stokes equations can further simpify to the Euler equations. Υποσημειώσεις Εσωτερική Αρθρογραφία * Φυσικός Νόμος * Εξίσωση *Εξισώσεις *Navier–Stokes equations *Computational fluid dynamics Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *[ ] *[ ] Κατηγορία: Νόμοι Ρευστομηχανικής Κατηγορία: Νόμοι Κλασσικής Φυσικής Κατηγορία:Νόμοι Φυσικής Συνεχούς Μέσου